Modelling the impact of interventions on imported, introduced and indigenous malaria infections in Zanzibar, Tanzania

Malaria cases can be classified as imported, introduced or indigenous cases. The World Health Organization’s definition of malaria elimination requires an area to demonstrate that no new indigenous cases have occurred in the last three years. Here, we present a stochastic metapopulation model of malaria transmission that distinguishes between imported, introduced and indigenous cases, and can be used to test the impact of new interventions in a setting with low transmission and ongoing case importation. We use human movement and malaria prevalence data from Zanzibar, Tanzania, to parameterise the model. We test increasing the coverage of interventions such as reactive case detection; implementing new interventions including reactive drug administration and treatment of infected travellers; and consider the potential impact of a reduction in transmission on Zanzibar and mainland Tanzania. We find that the majority of new cases on both major islands of Zanzibar are indigenous cases, despite high case importation rates. Combinations of interventions that increase the number of infections treated through reactive case detection or reactive drug administration can lead to substantial decreases in malaria incidence, but for elimination within the next 40 years, transmission reduction in both Zanzibar and mainland Tanzania is necessary.

Thus, the RCD term was modified to with ν (n) k either 20 or 100.

S1.1.4 Switching from RCD to RDA
When modelling RDA, we considered that all index household members and neighbours (when included) would receive treatment regardless of disease status. Thus, the diagnostic test sensitivity, ρ, was changed from 34% to 100%.

S1.1.5 Treatment of a proportion of cases brought on to Zanzibar by travelling humans (either residents or visitors)
Currently, prophylaxis is not provided to travellers to mainland Tanzania. Similarly, there is no screen-and-treat programme for entrants to Zanzibar. We include treatment of imported cases as a potential intervention in our model, in order to evaluate what proportion of cases must be treated to achieve different reductions in prevalence on Pemba and Unguja [2]. We modify Eq. (4) in the main text to have a θ outbound , which includes treatment for visitors from mainland Tanzania on their outbound journey to Zanzibar, and θ return for Zanzibari residents that receive treatment on their return journey to Zanzibar. Thus, the base form of the equation becomes O represents the proportion of travellers from mainland Tanzania receiving treatment such that they are no longer infected upon entering Zanzibar, and R represents the proportion of Zanzibari residents receiving treatment such that they are no longer infected upon returning to Zanzibar. We always simulate equal proportions of outbound and return cases being treated (i.e. O = R) S1. 1

.6 Reductions in the malaria transmission rate
The rate at which malaria is transmitted from one human to another can be reduced through vector control interventions such as the use of long-lasting insecticidal nets, indoor residual  Thus, for this intervention, β is replaced by β(1 − r) where r is the reduction in vectorial capacity. As vectorial capacity is proportional to the number of susceptible humans infected by an infected human per day, β is proportional to the vectorial capacity, and any reduction in β could arise from a proportional reduction in the vectorial capacity [3]. Values ranging from 0.25 to 0.9 were tested for r on Pemba and Unguja, and values ranging from 0.1 to 0.3 for r on mainland Tanzania.

S2.1 Comparison of models
The current model was compared to the simpler version of the model presented in Das et al (2022) [2]. The counterfactual increase in malaria prevalence expected from stopping RCD was found to be very similar in both cases (Table S1). The slight difference is due to the stochastic nature of the model implementation.

S2.2 Comparison of interventions
The impact of each intervention alone was tested by changing one factor at a time and plotting the final equilibrium reached 40 years after the introduction of the intervention. All other factors were held at their baseline value, given in Table 3 in the main text. The results from this analysis are shown in Fig S1. Most intervention parameters had an approximately linear relationship with malaria incidence, but the relationship between the percentage of travellers treated and the incidence of infections was mildly concave, and the relationship between a reduction in the malaria transmission rate in Zanzibar and the incidence of infections was steeply curved. This suggests that even small increases in vector control may have a disproportionately large impact with regard to reducing malaria incidence on Zanzibar.

S2.3 Impact of parameter uncertainty
Parameter uncertainty was considered in the same way as described in Das et al (2022) [2]. Simulations were run with a range of parameter values based on the uncertainty in the data, taking the posterior distribution when an uninformative prior is updated with the observed data.
The parameters varied and the distributions from which they were sampled were as follows: • The equilibrium malaria prevalence on Pemba, I * 1 ∼ Beta(32, 2242) ; • The equilibrium malaria prevalence on Unguja, I * 2 ∼ Beta(92, 3196);

S5
• The targeting ratio in index households in Pemba, τ ; • The number of people tested by the RCD programme in the index household in Pemba, ν 1 ∼ Normal(7.02, 0.24); • The number of people tested by the RCD programme in the index household in Unguja, ν 2 ∼ Normal(6.36, 0.25); • The number of people tested by the RCD programme in neighbouring households in Pemba, ν 1 ∼ Normal(20.36, 0.50); • The number of people tested by the RCD programme in neighbouring households in Unguja, ν  100 random values were selected from these parameter distributions, and each set of values was simulated with five different seeds, forming a total of 500 simulations for each intervention scenario. The final equilibrium value reached for a range of interventions, along with the uncertainty stemming from both the parameter and stochastic variation, is shown in Fig S2a. The distribution of annual incidence at equilibrium can be seen in Fig S2b. The impact of parameter uncertainty on the probability of reaching elimination was also examined and found to be minor (see Fig S3). Even when parameter uncertainty is included, elimination is only observed when there is 100% importation treatment in the absence of transmission reduction interventions.

S2.4 Sensitivity analysis
A sensitivity analysis was conducted using the Sobol method to characterise the impact of parameter variation on model outputs. 32,768 parameter values were sampled from uniform distributions for each parameter using Saltelli sampling [9,10]. The bounds of the uniform distributions corresponded to 95% confidence intervals found in the literature or, when such bounds were not available, the point estimate of the parameter ±50%. Upper and lower bounds and data sources can be found in This analysis suggests that the main outputs of malaria prevalence and the annual incidence of indigenous infections are most sensitive to the estimates of the transmission parameter (Fig S4  and S5). As these are back-calculated from the baseline malaria prevalence, the need for accurate estimates of the prevalence in the general population is important for a correct estimate of the effective human-to-human malaria transmission rate. An accurate estimate of the baseline prevalence is also key to estimating the probability of elimination being reached under different

First order indices Total indices
Prevalence on Pemba

First order indices Total indices
Prevalence on Zanzibar Figure S4: First order and total Sobol indices for each parameter tested when the overall malaria prevalence is considered as the output. Note, the 95% confidence intervals are smaller than the point sizes and so are not visible. The model output of malaria prevalence on Zanzibar as a whole was calculated by multiplying the expected prevalence on each island by the population size of each island, summing to get the total number of infected people on both islands, and then dividing by the summed population across both islands.  Figure S5: First order and total Sobol indices for each parameter tested when the annual incidence of indigenous cases is considered as the output. Note, the 95% confidence intervals are smaller than the point sizes and so are not visible. The total incidence of indigenous cases for Zanzibar as a whole was calculated by summing the incidence of indigenous cases on each island.

S2.5 The impact of a fixed versus a varying targeting ratio
The targeting ratio is calculated from the RADZEC study data and is assumed to be fixed in the main text, regardless of the population malaria prevalence. This implies that cases do not become more clustered as the disease prevalence falls. In comparison, Chitnis et al (2019) consider a targeting ratio that varies depending on prevalence and the number of people tested, with the ratio of malaria infections amongst those tested as compared to the general population decreasing as prevalence and the number of people tested increases [13]. The following function was found to best estimate the targeting ratio, τ , for geo-located prevalence data collected in Zambia: where τ is the targeting ratio, ν is the number of people tested, not including the index case, N is the total population, and α 1 , α 2 and α 3 are fitted parameters with values α 1 = 0.23 (95% credible interval (CI): 0.16, 0.29); α 2 = −1.40 (CI: -2.77, -0.02) and α 3 = 2.87 (CI: 1.13, 4.59) [13].
In order to compare running the model with a fixed targeting ratio and a varying targeting ratio, we take the function from Chitnis et al (2019) that is fitted to data from Zambia, and apply a scaling factor to adjust the targeting ratio so that the targeting ratio matches between Eq. (5) and the targeting ratio for the index household in the RADZEC data (τ (h) in the main text). Thus, the equation we used to generate a varying τ was given by where A was calculated to be 0.19 for Pemba and 0.42 for Unguja, in order to match the targeting ratios calculated by Eq. (6) and the targeting ratio seen in the RADZEC study data. Thus, as the malaria prevalence decreases due to the introduction of new interventions, the targeting ratio increases and the effectiveness of RCD increases.
Running the model with a varying τ and a fixed τ , we see that the difference in the targeting ratio is not substantial even when considering the maximum RCD interventions tested i.e. RDA with triple the usual treatment seeking rate and 100 neighbours included in treatment (see Fig S6). These interventions maximise the effect of the targeting ratio and so are the ones where we'd expect to see the largest difference between the blue and purple lines in Fig S6. When RCD finds and treats a lot of cases, a targeting ratio that improves as the prevalence falls can provide an optimistic outlook of potentially eliminating malaria earlier than when considering a fixed targeting ratio, which makes the more conservative assumption of no increase in case clustering as prevalence decreases. Nonetheless, given the difference is small, we have used a fixed targeting ratio for all simulations shown in the main text.

S2.6 The impact of varying the definition of malaria re-establishment
Currently, we consider a simulation to have reached elimination when three years have passed with zero incidence of indigenous cases. However, if an indigenous case appears after this three year period, we count this as malaria re-establishment and thus losing 'eliminated' status. In contrast, the World Health Organization defines the minimum indication of re-establishment of transmission as 'the occurrence of three or more indigenous malaria cases of the same species per year in the same focus, for three consecutive years' [14]. Since no country that has been certified as malaria-free has lost certification, we additionally modelled the impact of assuming that once a region eliminates malaria, it stays malaria-free. Out of the 500 simulations, when a simulation reaches three years with zero incidence of indigenous cases, we assume it remains at zero indigenous cases indefinitely into the future. The probability of reaching elimination is shown in Fig S7, with the assumption of remaining malaria-free after elimination labelled as 'cumulative' and the more strict definition of malaria re-establishment (losing 'eliminated' status after the appearance of one indigenous case) labelled as 'transient'. We observe that in the majority of cases, the definition of re-establishment does not impact the proportion of runs reaching elimination. Only in the case where the number of indigenous cases is typically zero, but not always (90% importation treatment with 90% reduction in the transmission rate on Pemba) does the definition of re-establishment make a substantial difference to the number of runs reaching elimination. Figure S11: Median incidence of infections from 500 stochastic simulations comparing the current reactive case detection (RCD) system with 100% follow up of cases and 100 neighbours being included in testing and treating to reactive drug administration (RDA; in reaction to detecting a case at a health facility, upon follow up, testing is skipped and antimalarials are given to all index household members and 100 neighbours) and increases in the rate at which people seek treatment (treatment seeking is abbreviated as 'TS').  Figure S15: Proportion of stochastic simulations reaching elimination (three years with zero indigenous cases) when 100% of infected travellers from mainland Tanzania are treated, starting from year 0. We assume that all other interventions are at baseline values (reactive case detection for 35% of cases arriving at a health facility at the index household level only).